Question

Salary Negotiation A welder's contract promises a 3.5\(\%\) salary increase each year for 4 years and Luisa has an initial salary of \(\$ 36,500\) . (a) Show that Luisa's salary is given by $$y=36,500(1.035)^{\text { int } t}$$ where \(t\) is the time, measured in years, since Luisa signed the contract. (b) Graph Luisa's salary function. At what values of \(t\) is it continuous?

Short answer

Luisa's salary is modeled by the function y = 36,500(1.035^t), where \(t\) is the time in years since the contract signing which is continuous for all \(t \geq 0\).

Step-by-step solution

Understanding the Mathematical Model

The salary of Luisa is modeled by an exponential function, where the initial salary of $36500 is multiplied by \(1.035^t\), and \(t\) is the number of years since the contract signing. The 1.035 in the equation represents a 3.5% raise each year (1+3.5/100=1.035).

Validation of the Mathematical Model

Replace y with 36,500 and t with 0, as per the initial conditions. The result is 36,500 = 36,500 * \(1.035^0\) which simplifies to 36,500 = 36,500 validating the initial condition. Thus, the provided salary model is accurate.

Sketching the Function y = 36500 * \(1.035^t\)

When graphing the function, we see it takes the form of an exponential growth curve starting at $36,500 when t = 0 (the point at which Luisa signed the contract) and increases every year due to the annual salary increase.

Continuity of the Function

The function y = 36,500(1.035^t) is continuous for all values of \(t\), because \(t\) is the time measured in years since signing the contract, and the domain of this exponential function is all real numbers. Therefore, it is continuous for all \(t \geq 0\).

Interpreting the Function

The function shows that Luisa's salary is continuously increasing each year by 3.5\%. This type of salary increase is common in many job contracts where a fixed percentage increase is applied each year.